Experience-dependent spatial growth model

In our model we have no concrete representation of the dendritic tree or the axonal branches. However, our neurons are initially retinotop organized and the connections between two layers are described by a set of pairs of presynaptic and postsynaptic neurons. Regard- ing a single postsynaptic neuron, this gives a weight matrix with the retinotop layout of the presynaptic layer. Thus, we can locate our synapses in the coordinate system of the afferent layer. Despite this, we lack a concrete representation of the position of the axonal-dentrite contact. However, we can make the likely assumption that connections to neighboring presynaptic neurons would be in immediate vicinity of the regarded synapse. Hence, we define the neighboring synapses in the retinotop grid as potential synapses, being likely to be formed in the case of a spine outgrowth. Further, it has been found that it is likely that new synapses are formed in the vicinity of existing strong synapses (Caroni et al., 2012; Harvey et al., 2008; De Roo et al., 2008). Thus, we exploit the spatial synapse organiza- tion of our model to determine a probability for creating a synaptic connection based on the strength of neighboring synapses.

New formed synapses are originated by thin spines (Knott et al., 2006). Such spines are likely to fast disappear (Yasumatsu et al., 2008), spines with larger volume are found more stable (Knott et al., 2006; Yasumatsu et al., 2008). New spines are subjected to normal synaptic plasticity promoting stabilization (Caroni et al., 2012; Holtmaat et al., 2005), which acts much faster than structural changes. Therefore, newly created synapses in our model begin with a small randomly chosen synaptic weight value (see 4.2.1). The value is determined with an expectation value equal to the value where synapses are eliminated with the half of the maximum delete probability. This process is based on foundations that new formed synapses having mostly similar synaptic volumes as synapses which are likely to disappear (e.g. Yasumatsu et al., 2008).

Whereas the formation of new synapses seems to depend on the synaptic strengths in the neighboring (Caroni et al., 2012; Harvey et al., 2008; De Roo et al., 2008), synapse elimination is closely related to the volume of a synaptic spine (Yasumatsu et al., 2008). Thin spines are more likely to disappear than thick ones, which can be stable for long pe- riods (Knott et al., 2006; Kasai et al., 2010; Yasumatsu et al., 2008). We relate the deletion probability of synapses to the weight value (see 4.2.2) (Fauth et al., 2015b; Yasumatsu et al., 2008; Butz et al., 2009b). Low weights are likely to be removed whereas high one are subjected to be very stable (Yasumatsu et al., 2008).

4.2 EXPERIENCE-DEPENDENT SPATIAL GROWTH MODEL

FIGURE 4.1: Build probabilities around ex- isting synapses. Here the sum of the neigh- boring weights, mainly determining the build probability, is illustrated. Values in gray shaded boxes denote the weight strength of existing synapses. White boxes denote non- existing synapses, having the sum of neigh- boring weights inside. The illustration uses a neighborhood distance d of one. Non-existing synapses surrounded by strong synapses will have higher build probabilities than others.

4.2.1 Synapse creation

First we define the so called build probability pbuild_j for a non-existing synapse j (Eqn.

4.1). This probability describes how likely it is that a synapse is created within a particular time interval (see Sec. 4.2.3). This is determined based on the sum of the synaptic weights

wi of the existing synapses i in the neighborhood B( j, d) (Fig. 4.1), normalized by the

maximum weight wk of the postsynaptic neuron’s weight matrix and divided by the size

of its neighborhood|B( j, d)|. The result is scaled by the constant cs. The neighborhood

B( j, d) is defined as set of all synapses around synapse j, within a range d in any dimension

of the population grid.

pbuild_j = cs· 1 |B( j, d)|·_{i∈B( j,d)}

∑

Regarding a very large population of afferent neurons and a sparse connection struc- ture, the calculation of the build probability for each non-existing synapse (connection) is computationally inefficient. Thus, the equation is transformed to calculate just values in the neighborhood of existing synapses (Eqn. 4.2). Therefore, we calculate the probability incrementally by accumulating the normalized weights of the existing synapses i onto the

non-existing synapses in their neighborhood. So we can write the probability pbuild_j as sum

of all synaptic weights in the neighborhood B( j, d), where each (∀i ∈ B( j, d)) synaptic

4 STRUCTURALPLASTICITY

0 0

weight strength

whalf 2 · whalf _max k(wk) delete probability apdelmax pdelmax 2 pdelmax

FIGURE 4.2: Delete probability function with parameters. The func- tion drops from its extreme value, the maximal delete probability pdelmax, close to zero. For weights with weight strength whal f _{the deletion probability}

is half of its maximum and for twice this weight strength its a times the maximal probability.

gible difference at the borders of a population, where the neighborhood sizes can differ.

pbuild_j = cs·

∑

With this definition at hand, we can calculate the pbuild values for all relevant non-

existing synapses by iterating over the existing synapses i and increasing the build proba- bilities in their neighborhood (Eqn. 4.3).

pbuild(B(i, d)) = pbuild(B(i, d)) + cs· 1 |B(i, d)|

wi max_k(wk)

∀i (4.3)

If a new synapse is created, its weight wjis chosen from a uniform distribution between

zero and twice the value whal f multiplied with the maximum weight maxk(wk) of the neu-

ron. This places the weight value around the value where the deletion probability is half its maximum. The used parameters are listed in Table A.1.

4.2.2 Synapse removal

As well as for the formation of synapses we treat the removal as probabilistic process. Con-

trarily, we base the probability pdel_i for deleting an existing synapse i only on the weight

strength wi of the synapse itself (Eqn. 4.4). The probability follows a logistic function,

having its maximum for low weight values to remove weak synapses with higher probabil-

ity (Fig. 4.2). If the normalized weight strength increases to whal f the removal probability

is decreased to the half of its maximal value pdelmax. For two times whal f the removal

4.2 EXPERIENCE-DEPENDENT SPATIAL GROWTH MODEL

probability drops to the fraction a of its maximum. Hence, with the parameters whal f and

athe shape of the removal probability function can be adjusted. Where whal f defines the

setpoint between high and low removal probabilities and a controls the steepness of the transition, i.e. for very low values we get a sharp transient and a smooth one for higher values. pdel_i = p delmax 1+ e wi maxk(wk)−whal f ∆ (4.4)

The parameter ∆ (Eqn. 4.5) is defined so that the parameter a in relation to whal f sets

the result of equation 4.4 for wi= 2 · whal f to a· pdel

max

.

∆ = w

hal f

ln 1_a− 1
(4.5)

The used parameters are listed in Table A.2.

4.2.3 Probability calculation

The probabilities for synapse creation or removal pbuild|del are defined for a fixed fine

grained interval of tbase= 1s to allow a very smooth transition of network structures. In-

deed, structural plasticity is a very slow process in comparison to synaptic or intrinsic plasticity. Hence, to save computational costs, we update the connection structures just

every ∆t= 20s. Note, one network step has 1ms. Thus, we have to calculate how likely it

is that a synapse is created or removed with the given probability after an interval of ∆t. The probability for the opposing event that a synapse is not changed (formed or re- moved) is given by one minus the build or removal probability. We define that a synapse is changing after ∆t if the synapse is changed in one of the base intervals since the last update. We ignore the possibility that deleted synapses can be recreated and vice versa, which is suitable for low formation and removal probabilities. Hence, we get that a synapse will not change if the synapse has not changed within each of the previous base intervals. Thus, a synapse is changing with the probability of the opposing event (Eqn. 4.6).

4 STRUCTURALPLASTICITY

P(synapse not changed) = 1 − pbuild|del

P(synapse change after ∆t) = 1 − P(synapse not changed)tbase∆t

pchange(∆t) = 1 −1− pbuild|del

∆t

tbase

(4.6)